Abstract
Theq-deformed vertex operators of Frenkel and Reshetikhin are studied in the framework of Kashiwara's crystal base theory. It is shown that the vertex operators preserve the crystal structure, and are naturally labeled by the global crystal base. As an application the one point functions are calculated for the associated elliptic RSOS models, following the scheme of Kang et al. developed for the trigonometric vertex models.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andrews, G.E., Baxter, R.J., Forrester, P.J.: Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities. J. Stat. Phys.35, 193–266 (1984)
Date, E., Jimbo, M., Kuniba, A., Miwa, T., Okado, M.: Exactly solvable SOS models II: Proof of the star-triangle relation and combinatorial identities. Adv. Stud. Pure Math.16, 17–122 (1988)
Jimbo, M., Miwa, T., Okado, M.: Solvable lattice models related to the vector representation of classical simple Lie algebras. Commun. Math. Phys.116, 507–525 (1988)
Jimbo, M., Kuniba, A., Miwa, T., Okado, M.: TheA (1) n face models face models. Commun. Math. Phys.119, 543–565 (1988)
Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. London: Academic 1982
Kang, S.-J., Kashiwara, M., Misra, K.C., Miwa, T., Nakashima, T., Nakayashiki, A.: Affine crystals and vertex models. Int. J. Mod. Phys. A. Z. Supplement, 449–484 (1992)
Kang, S.-J., Kashiwara, M., Misra, K.C., Miwa, T., Nakashima, T., Nakayashiki, A.: Perfect crystals of quantum affine Lie algebras. Duke Math. J.68, 499–607 (1992)
Kashiwara, M.: On crystal bases of theq-analog of universal enveloping algebras. Duke Math. J.63, 465–516 (1991)
Kashiwara, M.: Global crystal bases of quantum groups. RIMS preprint756 (1991)
Frenkel, I.B., Reshetikhin, N.Yu.: Quantum affine algebras and holonomicq-difference equations. Commun. Math. Phys.146, 1–60 (1992)
Tsuchiya, A., Kanie, Y.: Vertex operators in conformal field theory onP 1 and monodromy representations of braid group. Adv. Stud. Pure Math.16, 297–372 (1988)
Lusztig, G.: Canonical bases arising from quantized enveloping algebras II. Progr. Theoret. Phys. Supplement102, 175–201 (1990)
Jimbo, M., Misra, K.C., Miwa, T., Okado, M.: Combinatorics of representations of\(U_q (\widehat{\mathfrak{s}\mathfrak{l}}(n))\) atq=0. Commun. Math. Phys.136, 543–566 (1991)
Kac, V.G.: Infinite dimensional Lie algebras, 3rd ed.. Cambridge: Cambridge University Press 1990
Lusztig, G.: Quantum deformations of certain simple modules over enveloping algebras. Adv. in Math.70, 237–249 (1988)
Kac, V.G., Wakimoto, M.: Modular and conformal invariance constraints in representation theory of affine algebras. Adv. in Math.70, 156–236 (1988)
Date, E., Jimbo, M., Kuniba, A., Miwa, T., Okado, M.: Exactly solvable SOS models: Local height probabilities and theta function identities. Nucl. Phys. B290, [FS20, 231–273 (1987)
Jimbo, M., Miwa, T., Okado, M.: Local state probabilities of solvable lattice models: AnA (1) n-1 family family. Nucl. Phys. B300, [FS22], 74–108 (1988)
Date, E., Jimbo, M., Kuniba, A., Miwa, T., Okado, M.: One-dimensional configuration sums in vertex models and affine Lie algebra characters. Lett. Math. Phys.17, 69–77 (1989)
Author information
Authors and Affiliations
Additional information
Communicated by A. Jaffe
Dedicated to Huzihiro Araki
Rights and permissions
About this article
Cite this article
Date, E., Jimbo, M. & Okado, M. Crystal base andq-vertex operators. Commun.Math. Phys. 155, 47–69 (1993). https://doi.org/10.1007/BF02100049
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02100049